On certain equation related to derivations on standard operator algebras and semiprime rings

Sa？etak In this paper we prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let A(X) be a standard operator algebra on X and let L (X) be an algebra of all bounded linear operators on X. Suppose we have a linear mapping D: A(X) → L (X) satisfying the relation D(Am+n)=D(Am)An+AmD(An) for all A A(X) and some fixed integers m≥1,n≥1. In this case there exists B L (X), such that D(A)=AB-BA holds for all A F(X), where F (X) denotes the ideal of all finite rank operators in L (X). Besides, D(Am)=AmB-BAm is fulfilled for all A A(X)